Higher Order Derivatives

1
Product Quotient Rule

• Higher Order Derivatives

2
The Product Rule

• Theorem. Let f and g be differentiable functions.
  Then the derivative of the product fg is
• (fg) '(x) f(x) g '(x) g(x) f '(x)
• In other words, first times the derivative of the
  second plus second times the derivative of the
  first.

3
Using the Product Rule

• Example

4
Product Rule

• Example

5
Product Rule

• Example

6
Quotient Rule

• Theorem. Let f and g be differentiable functions.
  Then the derivative of the quotient f/g is

In other words, low d high minus high d low over
low low.
7
Quotient Rule

• Example
• Find the derivative of

8
Rewriting Before Differentiating

• Example

9
Derivatives of Trigonometric Functions
10
Proof of Derivative of Tangent

• Considering

11
Differentiating Trig Functions

• y x tan x
• y x sec x y x(sec x tan x) sec x (1)
• sec x (x tan x 1)

12
Differentiating Trigonometric Functions
13
Higher Order Derivatives
Applications Finding Acceleration Due to
Gravity Population Growth p. 125 problem 79 Any
time we are asked to find the rate at which a
rate is changing, this is a second derivative.